J. Sens. Sens. Syst., 10, 1–4, 2021 https://doi.org/10.5194/jsss-10-1-2021 © Author(s) 2021. This work is distributed under the Creative Commons Attribution 4.0 License.

Explaining to different audiences the new definition and experimental realizations of the kilogram

Joaquín Valdés Universidad Nacional de San Martín, 25 de Mayo y Francia, 1650 San Martín, Prov. Buenos Aires, Argentina Correspondence: Joaquín Valdés ([email protected])

Received: 29 September 2020 – Revised: 23 November 2020 – Accepted: 30 November 2020 – Published: 21 January 2021

Abstract. Different options were discussed before reaching the final agreement on the new definitions of the SI units, effective from 20 May 2019, especially with regard to the kilogram, now defined in terms of the nu- merical value of the Planck constant (h). Replacing the artefact definition of the kilogram with a new one based on the mass of a particle, or the atomic mass constant (mu), would have been preferable for ease of understand- ing, among other reasons. In this paper we discuss some limitations of teaching to different audiences what a kilogram is in the redefined International System of Units (SI), including realizations of the new definition.

1 Understanding a definition of the kilogram based for electrical metrology to bring into the SI the realizations on the Planck constant of the volt and the ohm, providing exact numerical values for the Josephson constant KJ = 2e/h and the von Klitzing 2 On 16 November 2018, the General Conference of Weights constant RK = h/e . The second argument was that h is a and Measures – CGPM (from the French Conférence constant more fundamental than the mass of an elementary Générale des Poids et Mesures) decided that, effective from particle. 20 May 2019, the International System of Units, the SI, is the system of units defined in terms of seven defining constants. 1.1 School level This is the result of three decades of progress in metrology (Stock et al., 2019). It allows the dream of realizing the units An evasive way to explain what a kilogram now is, avoiding at any time and place. The new definition of the kilogram mentioning the Planck constant, would be not to answer ex- fixing the value of the Planck constant abolished the previ- actly what the new definition says, explaining instead that ous one, which was referred to as an artefact. It allows the one of the possible ways of “realizing” the new kilogram realization of mass at any scale using different technologies. is by counting atoms. Most people may have some idea of The kilogram is no longer the mass of “the weight that was what an atom is, so they can imagine that by putting together in Paris”, but it is defined by taking the fixed numerical value a huge number of atoms one obtains a mass similar to that of h to be exactly 6.62607015 × 10−34 when expressed in of the old platinum iridium artefact that served as an interna- the unit joules per second. Now we have to explain what this tional prototype of the kilogram. So far the explanation could new kilogram is to wider audiences, including technicians become understood even by elementary school pupils. The and university graduates from the most diverse disciplines, following immediate question will be of course how to gather who do not necessarily know what the Planck constant h is. and count so many atoms. This paves the way for explaining This difficulty was already known from the very beginning the efforts made during the last 20 years in order to measure of discussions around the redefinition of the SI. The question the number of atoms existing inside special silicon spheres. of understandability was present in several important pub- In principle, this first answer, which avoids referring to the lications, at least since 2007, until shortly before the final Planck constant, is not incorrect, since this so-called silicon approval of the redefinition (Becker et al., 2007; Fletcher et route for the realization of an “atomic kilogram” and a sec- al., 2015). One prevailing argument to support the definition ond route using a special Kibble balance for the realization of of the kilogram based on a fixed value of h was the need an “electric kilogram” may provide both a link between the

Published by Copernicus Publications on behalf of the AMA Association for Sensor Technology. 2 J. Valdés: Explaining to different audiences the new definition and experimental realizations of the kilogram

Planck constant and a macroscopic mass (Stenger and Göbel, proaches assumed that incorporating hands-on learning tech- 2012). niques will make it easier to understand something so com- plex. 1.2 Higher education levels In 2015 two experimental educational proposals for the understanding of the kilogram definition based on funda- For a more advanced audience, possibly with university stud- mental constants were published: one for the electric kilo- ies, or advanced pre-university studies, deserving of address- gram (Chao et al., 2015) and a second for the atomic kilo- ing the new definition of the kilogram based on the Planck’s gram (Davis, 2015), both to 1% relative uncertainty or less. constant, more complex approaches are required. A friendly The first was a watt balance of LEGO blocks constructed at way of introducing the Planck constant h is to motivate the National Institute of Standards and Technology – NIST. this particular audience with the provocative question about It allows explaining how the value of a mechanical force whether anyone knows why a fluorescent tube emits light. is precisely given by electrical measurements. Nevertheless, Next we introduce Bohr’s atom model, the difference in en- because the Kibble balance connects classical mechanics to ergy 1E between two energy levels and the emission of quantum mechanics, addressing some equations of the quan- a photon with frequency ν, presenting Planck’s equation tum effects involved is inevitable. Using any version of di- 1E = hν. Once introduced h, the unavoidable next question dactic watt balances one may explain how to compensate for will be what the relationship of h to a mass m is. A possi- the mechanical power delivered by the gravitational attrac- 2 ble answer is to bring to light Einstein’s equation E = mc . tion of a mass with the electrical power of a moving elec- Linking Einstein’s equation with the mentioned Planck’s re- tromagnetic force, but not the relationship of the mass that lation may show the connection between m and h. It was is being weighed with the Planck constant h. In the own even presented in this way for a time on the BIPM website. words of the authors of the LEGO balance, “Unfortunately, Nevertheless, care should be taken when combining both it still requires some abstraction to explain how electrical equations. The mass m in Einstein’s equation refers to the power is related to the Planck constant via the Josephson ef- rest mass of an object. The frequency ν in Planck’s equation 2 × fect and the quantum Hall effect”. The product KJ RK is usually refers to photons emitted with that frequency. Be- needed, relating electrical voltage and resistance through the cause photons are massless particles, equating both energies Josephson constant KJ = 2e/h and the von Klitzing constant 2 to show the relationship between h and m requires particu- RK = h/e . lar consideration with respect to the mass to which m refers. The second is a rather simple experiment conceived by This mass m could be, for example, the change in mass 1m Richard Stephen Davis to explain in 2015 how the kilogram of the particle when it emits a photon of frequency ν. would be redefined in 2018 in terms of the Planck constant, A second answer to the question of the relationship of h which is closely linked to the Avogadro and atomic mass to a mass m may be given by describing the realization of an constants. After measuring the volume and mass of a high- electric kilogram using the so-called Kibble balance, previ- purity aluminium cube, he estimated the number N of alu- ously known as watt balance (Robinson and Schlamminger, minium atoms inside the cube, taking the edge dimension a 2016). Equating mechanical and electrical power, it relates of the aluminium unit cell as determined in 1955 by X-ray the weight of 1 kg of mass to an electrical force needed to diffraction. Next the atomic mass of aluminium ma(Al) was 12 suspend it. The combination of two quantum effects, the estimated, also ma( C), the atomic mass of carbon-12, and quantum Hall effect discovered in 1980 by Klaus von Klitz- the atomic mass constant mu. As the molar mass constant 12 −1 ing, in conjunction with the previously predicted Josephson Mu = M( C)/12 was exactly 1 g mol before the revision effect, enables electrical power to be measured in terms of of the SI, a value for the Avogadro constant NA = Mu/mu the Planck constant h and frequency. According to the expe- was calculated from the measurement of mu. Using the equa- rience gathered in our institute of metrology over more than tion for the ionization of a hydrogen-like atom, NA and h ap- 20 years of teaching the SI to students with basic knowledge pear linked themselves and with other constants known to in natural sciences, it is not required that they necessarily have very small uncertainty. Being mu = Mu/NA and NA re- have completed a course in quantum physics. The essence of lated to h, from the measured value of mu a value for h was both quantum effects may be explained without developing also obtained. That simple experiment allowed determining the results starting from the Schrödinger equation. the Avogadro constant and the Planck constant, both with an uncertainty of less than 1 %, in close analogy to the way the 2 Learning by doing: experimental proposals for most accurate experiments with silicon spheres achieved an teaching the new kilogram before the revision of uncertainty of a few parts in 108. the SI After comparing both experiments described, we found it more interesting and less expensive to implement the exper- Anticipating the important difficulties that would arise in iment with the high-purity aluminium cube, as proposed by teaching the kilogram to be redefined, experimental solutions Richard Davis, than a tabletop watt balance capable of mea- emerged already some years before its adoption. These ap- suring gram-level masses to 1 % relative uncertainty, like the

J. Sens. Sens. Syst., 10, 1–4, 2021 https://doi.org/10.5194/jsss-10-1-2021 J. Valdés: Explaining to different audiences the new definition and experimental realizations of the kilogram 3

LEGO balance quoted. With a homogenized aluminium cube (not free of imperfections and impurities) specially prepared by an Argentine manufacturer of electrolytic aluminium, the difference between the mass calculated counting atoms and the mass measured by weighing turned out to be 0.1 % or less. On the other hand, the experiment with the silicon cube, adapted to engineering students, makes it possible to simul- taneously implement dimensional and mass precision mea- surements of interest for teaching metrology at that level.

3 Explaining the realization of the new kilogram after the revision of the SI

Both experimental educational solutions for the realization of the kilogram mentioned in the previous section were devel- oped before the redefinition of the SI. The purpose of such experiments changed after the redefinition. For example, in the Kibble balance approach, the experiment was initially used to measure h from a mass traceable to the international prototype of the kilogram. After the redefinition, it is used Figure 1. Experimental determination to 0.1 % of the mass of an aluminium cube following the new definition of the kilogram trace- to realize the definition of the kilogram from the value of h able to the Planck constant. The cube volume is obtained by di- fixed by the CGPM in 2018. The values of the constants to mensional measurements (a), optionally also by hydrostatic weigh- be used before the redefinition may be also different after ing (b). The number of atoms N is calculated taking a precise value the redefinition. In the Kibble balance approach, for exam- of the edge dimension a of the aluminium unit cell. The mass of ple, the values of KJ and RK used in 2015 were conventional the cube is then expressed as mc = Nm(Al), being m(Al) the mass values agreed on for electrical measurements since 1990, ab- of a single aluminium atom, obtained from the quotient h/matom breviated as KJ-90 and RK-90. Both values changed with the provided to high accuracy by atom interferometry. redefinition of the SI. In the case of the experiment in a classroom setting with −10 the aluminium cube described, more conceptual changes at present were reported with 2.0 × 10 accuracy by Hol- need to be taken into account. As the molar mass constant ger Müller’s group at the University of California, Berkeley, 133 was exactly 1 g mol−1 before the revision of the SI, a value in h/m( Cs) measurements (Parker et al., 2018). for the Avogadro constant NA was calculated from the mea- If we want to use at present the didactic experiment of the surement of mu. However, after the redefinition of the SI, aluminium cube, it should no longer be used to determine the the molar mass constant is no longer exactly 1 g mol−1, and values of certain constants which have been fixed but rather NA is now fixed in the definition of the mole. to obtain the mass of the cube following the corresponding Along with the redefinition of the units, new ways of car- mise en pratique of the kilogram (Fig. 1). After calculating rying out their realizations came into effect. The so-called the number of atoms N it contains, we need to address a value mises en pratique are documents indicating how the defi- of h/matom, i.e. h/m(Al) in the case of aluminium. The sim- nition of an SI unit may be realized in practice. The last plicity of the experiment before the redefinition of the SI is version of the mises en pratique for the definition of the no longer such, having to explain now atom interferometry, kilogram dates from 4 August 2020 (BIPM, 2020). In the requiring knowledge of quantum physics and relativity. case of the silicon route for the realization of the so-called “atomic kilogram”, the mass ms of a sphere is first expressed 4 Conclusions in terms of the mass matom of a single atom: ms = Nmatom. N is the number of atoms in the sphere experimentally de- The previous definition of the kilogram referred to an artefact termined using X-ray crystal density methods. The mass of that involved the risk of loss, damage, or change. The cur- an atom expressed in kilograms is now obtained from the rent definition of the kilogram based on a constant of nature quotient matom/h. This ratio is provided to high accuracy solves the “artefact problem” and brings additional benefits. by atom interferometry, measuring the recoil of an atom For instance, it allows one to realize mass at any scale using absorbing a photon (Cladé et al., 2016). Now that the nu- different technologies. Additionally, having fixed the value of merical value of h is fixed, the mass of an atom may be the elementary charge e in the definition of the ampere, the known with the uncertainty of the experiments determining election of fixing the Planck constant h in the definition of the 2 matom/h with atom interferometry. The best available results kilogram, allowing values of KJ = 2e/h and RK = h/e with https://doi.org/10.5194/jsss-10-1-2021 J. Sens. Sens. Syst., 10, 1–4, 2021 4 J. Valdés: Explaining to different audiences the new definition and experimental realizations of the kilogram zero uncertainty, brings improvements in electrical metrol- Acknowledgements. The author thanks Jorge Sánchez for his ogy. However, a complete understanding of the new defini- commitment to the hands-on teaching experiments. tion of the kilogram will remain limited to an audience able to understand what Planck’s constant is, and its relationship to the mass of a body. Teaching the new mise en pratique of Review statement. This paper was edited by Klaus-Dieter Som- the kilogram is no easy task. The realization by comparing mer and reviewed by two anonymous referees. electrical power to mechanical power requires knowledge of two quantum effects, providing the Josephson constant KJ and the von Klitzing constant RK, even using a didactic Kib- References ble balance. The realization with the X-ray-crystal-density method (the silicon route) seems to be easier to explain. In Becker, P., De Bièvre, P., Fujii, K., Glaeser, M., Inglis, B., Lueb- principle, it comes from a classical idea where the mass of a big, H., and Mana, G.: Considerations on future redefinitions of the kilogram, the mole and of other units, Metrologia, 44, 1–14, pure substance can be expressed in terms of the number of el- https://doi.org/10.1088/0026-1394/44/1/001, 2007. ementary entities in the substance. Nevertheless, explaining BIPM: SI Brochure, 9th Edn., Appendix 2, 1–4, available at: https: that the SI value of atomic masses is obtained from atom in- //www.bipm.org/utils/en/pdf/si-mep/SI-App2-kilogram.pdf (last terferometry experiments also requires advanced knowledge access: 18 January 2021), 2020. of physics. Chao, L. S., Sclamminger, S., Newell, D. B., and Pratt, J. Although fixing the value of the atomic mass constant mu R.: A LEGO watt balance: An apparatus to determine a would have been preferable for ease of understanding, for mass based on the new SI, Am. J. Phys., 83, 913–922, other reasons it was decided to fix the value of the Planck https://doi.org/10.1119/1.4929898, 2015. constant h. The new SI allows linking measurements at the Cladé, P., Biraben, F., Julien, L., Nez, F., and Guellati-Khelifa, S.: atomic and quantum scales to those at the macroscopic level. Precise determination of the ratio h/mu: a way to link micro- The lower uncertainties now obtained for atomic mass val- scopic mass to the new kilogram, Metrologia, 53, A75–A82, https://doi.org/10.1088/0026-1394/53/5/A75, 2016. ues expressed in kilograms are still greater than those typi- Davis, R. S.: What Is a Kilogram in the Revised Interna- cally achieved in the comparison of atomic masses. For this tional System of Units (SI)?, J. Chem. Educ., 92, 1604–1609, reason, the atomic mass community will continue to use the https://doi.org/10.1021/acs.jchemed.5b00285, 2015. non-SI unit dalton instead of the kilogram. This scenario of Fletcher, N., Davis, R. S., Stock, M., and Milton, M. J. T.: Mod- living with two different units of mass may change if fu- ernizing the SI – implications of recent progress with the funda- ture upgrades for atom interferometry experiments increase mental constants, how scientists can derive the mass of a kilo- the accuracy by 2 orders of magnitude (Valdés, 2019). For gram from the fixed value of the Planck constant, available at: 140 years it was quite easy to teach what 1 kg is. The new https://arxiv.org/ftp/arxiv/papers/1510/1510.08324.pdf (last ac- definition based on Planck’s constant brings numerous ad- cess: 18 January 2021), 2015. vantages, but its understanding remains limited to a very re- Parker, R. H., Yu, C., Zhong, W., Estey, B., and Müller, H.: Mea- stricted audience. Several options have been presented to ex- surement of the fine-structure constant as a test of the standard model, Science, 360, 191–195, 2018. plain at different levels the new definition of the kilogram, Robinson, I. A. and Schlamminger, S.: The watt or Kibble balance: eventually using the realization based on counting atoms a technique for implementing the new SI definition of the unit of when the meaning of the Planck constant is not known. mass, Metrologia, 53, A46–A74, https://doi.org/10.1088/0026- 1394/53/5/A46, 2016. Stenger, J. and Göbel, E. O.: The silicon route to a primary realiza- Data availability. The data that support the findings of this study tion of the new kilogram (Letter to the Editor), Metrologia, 49, are available upon request to the author if required. L25–L27, https://doi.org/10.1088/0026-1394/49/6/L25, 2012. Stock, M., Davis, R., de Mirandés, E., and Miltom, M. J. T.: The revision of the SI - the result of three decades of progress in Competing interests. The author declares that he has no conflict metrology, Metrologia, 56, 02200, https://doi.org/10.1088/1681- of interest. 7575/ab0013, 2019. Valdés, J.: Reviewing the Revised International System of Units (SI), Elsevier, Adv. Imag. Elect. Phys., 211, 121–186, Special issue statement. This article is part of the special issue https://doi.org/10.1016/bs.aiep.2019.05.001, 2019. “Sensors and Measurement Science International SMSI 2020”. It is a result of the Sensor and Measurement Science International, Nuremberg, Germany, 22–25 June 2020.

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